section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2exOPEs from Poincaré Symmetry

Throughout, we take \(z\)  and \(\bar{z}\)  to be independent and continue the Lorentz symmetry \({\rm SL}(2, \mathbb {C})\)  to \({\rm SL}(2, \mathbb {R})_L \otimes {\rm SL}(2, \mathbb {R})_R\) , following for example [1]}, [2]}. Working to leading order in a holomorphic limit \(z \rightarrow 0\)  with \({\bar{z}}\)  fixed, the contribution from primary operators in the celestial OPE at tree-level has been demonstrated [1]} to take the formIt would be interesting to provide a general derivation of the leading holomorphic behavior from symmetry, as given in [4]}, [5]} for the MHV sector of gauge theory and gravity.
\( \begin{split}\mathcal {O}_{h_1, {\bar{h}}_1}(z, {\bar{z}})\mathcal {O}_{h_2,{\bar{h}}_2}(0,0)\sim \frac{1}{z}\sum _{p} C_{p} ( {\bar{h}}_1, {\bar{h}}_2){\bar{z}}^{p}\mathcal {O}_{h_1+h_2-1, {\bar{h}}_1+{\bar{h}}_2+p}(0, 0).\end{split}\) 
